Inside the inverted proofs class: Design challenges

January 28, 2013, 7:45 am

This is the second post in a series on the nuts and bolts behind the inverted transition-to-proofs course. The first post addressed the reasons why I decided to turn the course from quasi-inverted to fully inverted. Over the next two posts, I’m going to get into the design of the course and some of the principles I kept in mind both before and during the semester to help make the course work. Here I want to talk about some of the design challenges we face when thinking about MTH 210.

As with most courses, I wanted to begin with the end in mind. Before the semester begins, when I think about how the semester will end, the basic questions for me are: What do I want students to be able to do, and how should they be doing it?

This course has a fairly well defined, standard set of objectives, all centered around using logic and writing mathematical proofs. I made up this list that has all those objectives in one place, arranged by section in the book use. What always strikes me about this list is not so much is sheer size – which is daunting enough – but its diversity. We expect mastery of what you might call “mechanical” skills like writing the contrapositive of a conditional statement or doing modular arithmetic. But we also want students to master some high-level conceptual tasks like evaluating the validity of a proof or doing investigations to come up with conjectures. And of course, students must be able to write correct proofs of mathematical statements.

On top of these concrete objectives are metacognitive objectives that I hold for my versions of MTH 210. These have more to do with what I want students to be like at the end of the course. I visualize for MTH 210 students who are coinfident, competent. independent, creative, persistent problem solvers. I want to see students who are not only competent at the math we do in MTH 210, but who also like it.

No pressure, right? And it gets better, because mixed in with all these goals and aspirations is the plain fact that a lot of students have never seen math like this before and are in for a seismic culture shock. As I mentioned last time, many students coming out of a year of calculus (the prerequisite for this course) have a conception that mathematics = computation, specifically hand computation using basic algorithms, and mathematics is the study of completing these calculations with maximum efficiency and correctness. Proofs, by contrast, were those strange things you glossed over in geometry and trig. Many of them have literally over a decade of school math to reinforce this conception, a time frame that unfortunately usually includes the calculus experience they just completed.

So in other words, this course has a lot of challenges. It’s broad. It’s deep. It’s rigorous. It has a huge workload for a three-credit class. (Did I mention it was just a three-credit class?) And not only all that, but it goes completely against what many of the students think mathematics even is. There are intellectual mountains to climb as well as cultural and even emotional ones.

Let me say a bit about that last statement regarding emotions. Students can get really attached to the math-equals-computation model. For many of them, “math” — as they’ve come to know it — is the only place they ever felt at home in school. They’ve gotten really good at doing the rote mechanics that their teachers have asked them to do, and factoring polynomials, solving quadratic equations, doing derivatives — for some, these basic mechanical skills represent their identities as students. This self-referential web of hand calculation techniques represents a safe intellectual place for many new math majors, maybe the only such place they have. And here I come, telling them that this is not What Mathematics Is All About™ and pushing them out of that safety zone. This adjustment — learning what mathematics really is all about and gaining some facility with it — must happen if the students are to proceed further with their study of the subject. But it would be foolish to forget how deeply upsetting this can be. This is something you have to design around.

There’s one final set of constraints to keep in mind. At my university, all students take a semester of writing and then go on to take two “Supplemental Writing Skills” (SWS) courses, at least one outside their home discipline. MTH 210 is the designated SWS course. Among the things that means is that 1/3 of the students’ grade in MTH 210 must come from writing assignments that involve multiple submissions of drafts and revisions. The standard such assignment for MTH 210 is the Proof Portfolio.

The Proof Portfolio is a sort of semester-long project where students solve between eight and ten challenging mathematical problems that are proof-oriented. Here’s one from my portfolio assignment from Fall 2011:

Make a conjecture about a formula for the product
\[ \left( 1 - \frac{1}{4}\right) \cdot \left( 1 - \frac{1}{9}\right) \cdot \left( 1 - \frac{1}{16}\right) \cdot \cdots \cdot \left( 1 - \frac{1}{n^2}\right) \]
for all natural numbers \(n\) with \(n \geq 2\). Then, state your conjecture as a proposition and prove it.

I assign these problems in groups, usually three at a time, with the first group coming in the second week of the semester and the remaining groups spaced out about 3 weeks apart. So as students learn the math and proof techniques, they are getting problems to work that put those ideas to work. (I’d rate the above around a “5” on a scale of 10 for dificulty. It’s a standard mathematical induction proof, but the trick is that students have to discover what the theorem is, and then prove it.) Students submit up to two preliminary drafts for each problem during the semester, and then a final draft at the end. When I get a draft, I comment on it and send comments back to the students, who then revise their work.

If that sounds like a lot of grading — you don’t know the half of it. I’ll talk about that more in an upcoming post. But suffice to say it’s a lot of work for both me and for the students. And yet it’s a fundamental element of all MTH 210 offerings at my university. I have freedom to tinker with this assignment, but the expectation is that my classes will have a Proof Portfolio in them. This, too, is something to design around.

That gives a pretty complete picture of the landscape of the course, prior to making actual design decisions. In the next post, I’ll discuss what those decisions are, and how I used the inverted classroom concept to try to build all these constraints and conditions into the course.


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